Joys of Haar Measure

From the earliest days of measure theory, invariant measures have held the interests of geometers and analysts alike, with the Haar measure playing an especially delightful role. The aim of this book is to present invariant measures on topological groups, progressing from special cases to the more general. Presenting existence proofs in special cases, such as compact metrizable groups, highlights how the added assumptions give insight into just what the Haar measure is like; tools from different aspects of analysis and/or combinatorics demonstrate the diverse views afforded the subject. After presenting the compact case, applications indicate how these tools can find use. The generalisation to locally compact groups is then presented and applied to show relations between metric and measure theoretic invariance. Steinlage's approach to the general problem of homogeneous action in the locally compact setting shows how Banach's approach and that of Cartan and Weil can be unified with good effect. Finally, the situation of a nonlocally compact Polish group is discussed briefly with the surprisingly unsettling consequences indicated. The book is accessible to graduate and advanced undergraduate students who have been exposed to a basic course in real variables, although the authors do review the development of the Lebesgue measure. It will be a stimulating reference for students and professors who use the Haar measure in their studies and research.

The book under review is a serious adventure in Measure Theory and not for the faint of heart. It is a wonderfully structured guide to some deep exploration of continuous quantity. At the heart of its exposition is the question of what it means to assign lengths, areas, volumes and hyper-volumes to abstract sets sitting somewhere in n-dimensional space." - MAA Online"The text under review, as the title suggests, is a somewhat unorthodoxly written exposition of the theory of Haar measure. The joyous manner in which the authors present the material is unique and original, and quite catchy. Under the skillful hands of the authors, the subject, which is already inherently beautiful and elegant, comes to life in a very entertaining fashion. The book is well aimed at the graduate level, both with respect to the level of detail given and with respect to the expected mathematical maturity at that stage... The exposition is filled with historical detail and, at times, different mathematical points-of-view, leading to a pleasantly well-rounded understanding of the material and an appreciation of certain aspects of its development. I find the book to be particularly well suited as a self-study book for the motivated student, as well as a supplementary book for any course on the topic." - Zentralblatt Math

Joe Diestel, Kent State University, Kent, OH, USAAngela Spalsbury, Youngstown State University, Youngstown, OH, USA